Long-range superharmonic Josephson current and spin-triplet pairing correlations in a junction with ferromagnetic bilayers

The long-range spin-triplet supercurrent transport is an interesting phenomenon in the superconductor/ferromagnet ($S/F$) heterostructure containing noncollinear magnetic domains. Here we study the long-range superharmonic Josephson current in asymmetric $S/F_1/F_2/S$ junctions. It is demonstrated that this current is induced by spin-triplet pairs $\mid\uparrow\uparrow\rangle-$$\mid\downarrow\downarrow\rangle$ or $\mid\uparrow\uparrow\rangle+$$\mid\downarrow\downarrow\rangle$ in the thick $F_1$ layer. The magnetic rotation of the particularly thin $F_2$ layer will not only modulate the amplitude of the superharmonic current but also realise the conversion between $\mid\uparrow\uparrow\rangle-$$\mid\downarrow\downarrow\rangle$ and $\mid\uparrow\uparrow\rangle+$$\mid\downarrow\downarrow\rangle$. Moreover, the critical current shows an oscillatory dependence on thickness and exchange field in the $F_2$ layer. These effect can be used for engineering cryoelectronic devices manipulating the superharmonic current. In contrast, the critical current declines monotonically with increasing exchange field of the $F_1$ layer, and if the $F_1$ layer is converted into half-metal, the long-range supercurrent is prohibited but $\mid\uparrow\uparrow\rangle$ still exists within the entire $F_1$ region. This phenomenon contradicts the conventional wisdom and indicates the occurrence of spin and charge separation in present junction, which could lead to useful spintronics devices.